In the seminal paper [1]}, Witten laid the foundation of a three-dimensional topological quantum field theory, now called the Witten-Reshetikhin-Turaev TQFT. Implicit in this TQFT is a finite-dimensional projective linear representation of \(\Gamma _g\) , for any positive integer \(k\) , which is called the quantum representation at level \(k\) . These representations provide a powerful tool for studying the mapping class groups. Indeed, the collection of representations can distinguish any two mapping classes, except the identity and the hyper-elliptic involution in genus 2. This is the content of Turaev's asymptotic faithfulness conjecture, which was proved by Andersen using the theory of Toeplitz operators [2]}, [3]}.
By applying similar technology, the quantum representations were also used to prove that \(\Gamma _g\)  does not have Kazhdan's property (T) [4]}. Finally, the quantum representations have been used by Masbaum and Reid to show that any finite group occurs as a quotient of a finite-index subgroup of \(\Gamma _g\)  [5]}.
